Main readings and sources
McDL: Ch 12, plus ‘preference relations’
AUT: Lecture 3 - Axioms of Consumer Preference and the Theory of Choice (note, slightly different notation)
NS: Ch 2
Planned syllabus, coverage:
Plans: ‘(*)’: partially covered in pre-reqs
Key goals of this lecture (and accompanying self-study)…
Dr. David Reinstein, davidreinstein.wordpress.com
Office hours: Tuesday and Thursday 11am-noon during Spring term (just come by); Streatham Court 1.39 unless otherwise mentioned. or by appointment calendly.com/daaronr/20min/.
My research interests
My teaching and projects
Enrichment texts
Have a look at in case you are thinking of going for a PhD in economics or taking further coursework at the MRes level
Microeconomic Theory: Basic Principles and Extension (also by Nicholson and Snyder): more maths and extensions
David Autor’s MIT Open Courseware “Microeconomic Theory and Public Policy”: More rigorous, close to PhD level, includes good research applications; we’ll be drawing from it
PhD level: Jehle and Reny’s “Advanced Microeconomic Theory”, or Hal Varian’s text “Microeconomic Analysis”; Mas-Colell et al for wizards only
1. Economic basics (weeks 1-2)
Economic models, maths tools, introduction (NS ch. 1)
Utility, preferences, indifference curves, budget constraints (NS 2)
we build it up and then burn it down…
2. Build the model, put it together, examine it (weeks 3-5)
The Demand side:
Demand curves: Individual and market demand (NS ch. 3)
Profit maximisation and supply, perfect competition in a single market
Supply curves, entry/exit, Consumer and Producer Surplus, general equilibrium and welfare (brief)
3. How the market can go wrong (and how to maybe fix it) (weeks 6-7)
Market failures – Public goods
Monopolies; price discrimination as an imperfect remedy
4. Extensions to the model and applications (weeks 8-11)
Uncertainty (basic concepts, EU, risk aversion, investment choices)
Game theory; experimental evidence
`Behavioural’: Limits to cognition, willpower, self-interest; applications and evidence
What’s gonna be on the exam??
Examples of exam material: Practice problems in notes/lectures, problem sets, mock and sample questions on the VLE; mock exams, previous exams (esp since 2015-16)
##But anything I say you can find online, or in a book, so why are you here?
To interact
Not to hear what the *lecturer has to say, but for the lecturer to respond to what
The VLE and other resources
VLE let’s see it
Ask questions and make comments on the forum
LINK, I will monitor it
I may not have time to give a detailed answer to all individual emails; but you are welcome to come to my office hours –>
For now, available to you exclusively on the VLE
Contains all the lecture slide material and more, html links, etc.
In-lecture experiments and games
Ask me questions throughout the lecture
raise a white handkerchief if you are lost
Draw-along and solve-along
I will hot and cold-call
Interact and discuss: Peers, Forum, office hours
Interpret the exam instructions correctly! (and use your time wisely)
–>
Economists do not know everything (but we have thought through many arguments)}
Most non-economists do not fully understand these arguments, and they make mistakes, and they worry.
… more deep truths to follow, enough for now
Markets work well but not perfectly.
Imperfections in existing markets \(\rightarrow\) opportunities.
\(\rightarrow\) ‘All you can eat’ \(\rightarrow\) Spotify, Netflix, Kindle Unlimited
Imperfection: Lack of information about ‘experience goods’, lack of trust in one-shot-interactions
\(\rightarrow\) ??
\(\rightarrow\) ??
\(\rightarrow\) Uber, AirBnb, ‘bilateral reputation systems’
Shyness and fear of ‘losing face’
Ties in to my research …
‘Economics is the study of the allocation of scarce resources among alternative uses.’
‘Economics is the study of mankind in the ordinary business of life.’ Alfred Marshall
The study of the (economic) choices individuals and firms make and how these choices create markets.
Largely, using theoretical and mathematical ‘models’ that depend on strong assumptions.
The tortoise and the hare
What do models give us?
There are different views of this
Assumptions \(\rightarrow\) Results
and sometimes \(\rightarrow\) testable predictions (if the assumptions hold)
So why learn these models?
A starting point
(Sometimes) make testable concrete predictions
Building insight, clear arguments, a way of thinking
Discussion is framed around them; seen as a ‘baseline’
Understand the models to effectively critique or extend them
Differing views on the use of economic models
Instrumentalist:
The Methodology of Positive Economics (Friedman): the ultimate goal of theory is to “yield valid and meaningful … predictions about phenomena not yet observed”
‘Fictionalist’ (Sugden):
describes a fictional world that is credible or truthlike in something like the way that the events of a realistic novel are; the model connects with the real world by relations of similarity
Are these models predictive?
If not, are they useful?
Principle 1: Scarce Resources
Principle 2: Scarcity involves opportunity cost.
Above PPF: opportunity cost of more clothing is less food.
Principle 3: Opportunity costs are (often) increasing
See notes here
Studies of honeybees have found that they generally do not gather all of the nectar in a particular flower before moving on.
Why not?
Application 1.2 in NS text. Handout: some articles discussing this. Read at home, discuss
Consider the same for the UK/Exeter; give your best estimate
How does the analysis differ from the one your uncle would do?
Hey, ma and pa, what determines the price of a bread and the amount that gets sold?
… Describes how a good’s price and the quantity exchanged are determined
Adam Smith and the Invisible Hand
Prices
Not random nor morally determined
Signals to direct resources, reflecting the ‘worth’ of goods
Labour-cost-based theory of prices
Claim: If it takes twice as long for a hunter to catch a deer as to catch a beaver, one deer should trade for two beavers.
Why/when should this not hold (or not tell the whole story)?
Farmland was expanding in England … as new and less fertile land was brought into use,
it would naturally take more labor … to produce an extra bushel of grain.
Ricardo - Diminishing returns: the cost of producing one more of good A (here, food)—in other goods foregone—rises as more of A is produced
Diminishing returns/increasing costs:
Upwards-sloping supply curve, but where on this curve do we end up?
Argued price must equal both the value (to consumers) and the cost (to produce) of the last unit produced and consumed
Introduced the ‘demand curve’; with a downward slope – because:
‘Satiation’ (later units valued less) and catering to less keen consumers
With ‘single crossing’ there is a unique price where \(Q_s(p)=Q_d(p)\)
and a unique quantity where the last unit’s value to the consumer equals it’s cost to produce.
Draw: the famous Marshallian cross
Can you explain?
The inefficiency of any price other than where \(Q^D(p)=Q^S(p)\)?
If the price was set at a different value, what forces might push it to the equilibrium?
Who gains and who suffers with a government-imposed price floor/ceiling?
Do US farm subsidies help or hurt Africans in net?
Consider the effects on African farmers and African producers. How could we consider the `net effect’?
Some questions/problems I liked are in the handout… also see Problem Set 1 on the VLE
Be sure that you can do problems like 1.1 in the N&S text without difficulty;
For 1.1.C, note that for the supply curve, quantity supplied is never negative – below a certain price, it will just be zero.
Also consider ‘review questions’ 6 and 8 from the text
will go over the key parts of these problems, and you can ask questions
Math tools you must know – see handout, referring to NS text
(Univariate) functions, linear/nonlinear functions; the slope of a function (arc vs. point slope), concave/convex functions
Derivative of a function: a function that tells you the slope at each point; Minima, maxima
Functions of two or more variables, contour lines
(Simple) simultaneous equations
Slides, resources to help you, plus supplementary videos; www.khanacademy.org/math/
Lecture skips to Mini-lecture: Empirical microeconomics/econometrics here
Goals of this lecture (and accompanying self-study)
Overview of (re)-aquaintance with maths tools we will use
Flavour of what empirical microeconomics is, key issues in empirical work
Covers:
Nicholson/Snyder Chapter 1a: Mathematics used in Microeconomics
A very good resource – Khan academy: https://www.khanacademy.org/math/
##Simple stuff
Nonlinear (univariate) function : A function \(f(x)\) of a form other than \(f(x) = y=a+bX\);
For linear functions the slope is the same at any point. For nonlinear functions it may differ at each point.
E.g., the slope at \(x=1\) is $f’(x;x=1) = 1*1 - 4 = -2
The slope is zero where \(f'(x)=2x-4=0\), or where \(x=2\)
Oversimplifying:
Utility, profit, cost, production, returns, etc.
\[y=f(x,z)\]
\(y\) may increase and/or decrease in \(x\) and in \(z\),
The rate of increase of y in \(x\) may depend on the values of \(x\) and \(z\)
E.g., \[y=\sqrt(xz) = x^{1/2}z^{1/2}, x \geq 0, z \geq 0\]
Projecting a function up from X,Y space into the Z axis:
Level sets: E.g., indifference curves, isoquants and isocost curves.
Contour lines on a map
Consider a production function:
\[Y = f(K,L) = \sqrt(KL)\]
Setting this equal to 1 we can map out ‘all combinations of K and L that produce output \(Y=1\)’:
\[ Y = \sqrt(KL) = 1 \rightarrow KL = 1 \]
\[ \rightarrow K = 1/L \]
Setting this at Y = 2
\[ Y = \sqrt(KL) = 2 \rightarrow KL = 4 \] \[ \rightarrow K = 4/L \]
E.g.,
\[ X + Y = 3 \] \[ X - Y = 1 \]
Holds only where \(X=2, Y=1\), the ‘unique solution’
Meaningless to ask ‘how does a change in X affect Y?’ in the above context.
Uses evidence from the real world, i.e., observation, to answer questions
See, e.g.,
Trying to estimate demand curve, hypothesize linear function \[Q_d = a-bp\]
Suppose we know price is shifting because of costs, shifts in supply curve, or the firm experimenting
Observe price & quantity data for a period where ceteris paribus is reasonable
Fit ‘best’ line (minimise error) through these points
Estimate demand slope & intercept, use to make inferences
Never fits exactly. why not?
All [most] economic theories employ the assumption that ‘other things are held constant.’
the points may lie on several different demand curves, and attempting to force them into a single curve would be a mistake.
\(\rightarrow\) Carefully ‘control’ for other observable factors (a partial solution)
Suppose a model of supply and demand with ‘exogenous shifters’:
Now write the supply/demand model as: \[Q=-P+W\] \[P=Q+Z\]
Empirical work has estimated:
\[ Q = 85 - 0.4P \: (D) \] \[ Q = 55 + 0.6P \: (S) \]
Solving: \(85 - 0.4P=55+0.6P \rightarrow P = 30, Q = 73\)
Approximates 2000-2002 price
Why did price rise to US$130 in 2008 and fall to $50 by March ’09?
\[Q_D = 112 - 0.4P\] \[Q_S = 55 + 0.6P\]
\(\rightarrow\) solves to \(P=57, Q=87\)
##First problem set: coverage
See the Problem set 1 file on the VLE
Representative answers for each problem set given about 1-week after posting
Five support classes (tutorials), cover parts of these
E.g., …
E.g., “True or false: It is valid to plot observed prices and quantities traded in a market and fit a line through them to estimate a market demand curve.”
Consider a decision you recently made?
Define this decision clearly.
How do you think you decided among these options?
2 minutes: discuss with your neighbour
Suppose I asked you
‘State a rule that governs (determines/characterizes) how people do make decisions’…
I want this rule to be…
Informative (it rules out at least some sets of choices)
Predictive (people rarely if ever violate this rule)
Similar question:
‘State a rule that governs how people should make decisions’..
By ‘should’ I mean that they will not regret having made decisions in this way.
2 minutes: discuss with your neighbour
If people did follow these rules, what would this imply and predict?
Rules defined as ‘axioms about preferences’
‘Standard axioms’ \(\rightarrow\) (imply that) choices can be expressed by ‘individuals maximising utility functions subject to their budget constraints’
\(\rightarrow\) yields predictions for individual behavior, markets, etc.
Learn, understand, be able to explain and explain:
‘Utility’, how it’s defined
Key assumptions about preferences/choices; their implications
<!---
price ratios \& mgnl rates of substn.
-->
McDL: Ch 12, plus ‘preference relations’
AUT: Lecture 3 - Axioms of Consumer Preference and the Theory of Choice (note, slightly different notation)
NS: Ch 2
David Autor’s notes – MIT Microeconomic Theory and Public Policy Open CourseWare
Alt: The thing that people maximise when making economic decisions
How is this used?
Essentially, economists assume that `when making a choice among all available and feasible options, an individual will choose the one that yields the greatest utility’
\[Utility = U(X,Y; other)\]
Utility is not ‘observable and measurable in utils’
(Unlike midi-chlorians or thetans)
Revealed preference: if Al buys a cat instead of a dog, and a dog was cheaper, we assume Al gets more utility from a cat
Interpersonal comparisons are difficult
…Interpersonal comparisons are difficult
Standard utility functions are only ‘identified’ (only make distinct predictions) ‘up to a monotonic transformation’
i.e., a ‘rank preserving’ transformation.
E.g., multiply by a positive number, log, square, add/subtract from it;
\[A \succ B, B \succ A, \: or \: A \sim B \]
Fancy notation: Either A preferred to B, B preferred to A, or A indifferent to B
Forbidden: “I can’t choose between a ski holiday and a beach holiday, but I am not indifferent”
\[ A \succ B \: and \: B \succ C \rightarrow A \succ C \]
If I prefer an Apple to a Banana and a Banana to Cherry
then I prefer an Apple to a Cherry.
If not \(\rightarrow\) money pump.
Draw this!
ootnotesize{If the product is a ‘bad’ (e.g., pollution), redefine as the absence of the product}
If people obey the first two assumptions (axioms),\(^\ast\) they will make choices in a way consistent with maximising a (continuous) utility function
*(\(\ast\) and also ‘continuity’)
Continuity and nonsatiation
\(\rightarrow\) Compare utilities, depict using Indifference Curves
Formally (for 2 goods), the set of pairs of \(\{X,Y\}\) such that \(U(X,Y)=c\) for some constant \(c\)
Credit: www2.econ.iastate.edu
Credit: Frank’s Economics on the web (MIT)
Rank order of preference/indifference between points A-E.
Q1: How do we know \(E \succ B\) ?
Q: How do we know \(E \succ A\) ?
Why ‘voluntary trade’?
The indifference curve offers some intuition.
MRS = Absolute value of slope of indifference curve
‘Rate at which you’re willing to forgo consuming (\(Y\)) to consume one more (\(X\))’
(So the MRS is the ratio itself, because it’s the absolute value.)
Formally, the MRS at point \(X,Y\), where \(U(X,Y)=U_1\) is:
\[MRS(X,Y)=-\frac{dY}{dX}|U(X,Y)=U_1\]
Back to fig 2.2 (board or visualiser)
\(\rightarrow\) slope \(-2\), \(MRS=2\)
…willing to give up 1 hamburger to get 1 more soda \(\rightarrow MRS =1\)
\(MRS = \frac{1}{2}\)
More formal concept: (strictly) convex preferences \(\Leftrightarrow\) (strictly) ‘quasiconcave’ utility function, allowing ‘n goods’:
With diminishing MRS, for a ‘convex combination’ of the two bundles \(\mathbf{A}=(X_A,Y_A)\) and \(\mathbf{D}=(X_D,Y_D)\)
\[U(\alpha\mathbf{A}+(1-\alpha)\mathbf{D})\geq\alpha U(\mathbf{A}) + (1-\alpha)U(\mathbf{D})\]
where \(0<\alpha<1\).
(Note switch to a ‘weak inequality’ here for quasiconcavity, strict for ‘strict quasiconcavity’)
e.g., (with two goods),
\[U(\alpha X_A + (1-\alpha)X_D,\alpha Y_A + (1-\alpha)Y_D)\geq \] \[\alpha(U((X_A,Y_A)) + (1-\alpha)(U(X_D,Y_D))\]
Indifference curves never cross! And are never upwards sloping!
Goods A and B are Perfect Substitutes when an individual’s utility is linear in these goods
when she is always willing to trade off A for B at a fixed rate (not necessarily 1 for 1)
Goods A and B are Perfect Complements when an individual only gains utility from (more) A if she also consumes a defined (additional) amount of B, and vice-versa
These goods are ‘enjoyed only in fixed proportions’.
You cannot spend more than your (lifetime) income/wealth \(\rightarrow\) budget constraint.
If I spend all my income (I will do over a ‘relevant lifetime’):
Expenditure on X + Expenditure on Y = Income (I)
\[P_X X + P_Y Y = I \]
To see how \(Y\) trades off against \(X\), rearrange this to:
\[Y = -\frac{P_X}{P_Y} X + \frac{I}{P_Y}\]
If slope budget constraint = slope indifc curve at point X,Y \(\rightarrow\) \[P_X/P_Y = MRS(X,Y)\]
Warning: This equality holds at an optimal choice; it doesn’t hold everywhere.
At an optimal consumption choice (given above assumptions)
Consume all of income; locate on budget line; follows from ‘more is better’
Psychic tradeoff (MRS) equals market tradeoff (\(P_X/P_Y\)) if consuming both goods
Psychic tradeoff (MRS) equals market tradeoff (\(P_X/P_Y\)) if consuming both goods
Key intuition:
If I can give up X for (buy less X, get more Y) at some rate, and the benefit I get from doing this is at a different rate,
…then I can make myself better off.
Thus the original point could not have been optimal.
Recall \(U=U(X,Y)\).
\[U_X(X,Y) := MU_X(X,Y)\]
Derivative w/ respect to X: rate utility increases if we add a little X, holding Y constant
MRS: ‘how much Y would I be willing to give up to get a unit of X’?
Ans: Depends on marginal benefit of each … we can show \(MRS(X,Y)=\frac{MU_{X}}{MU_{Y}}\)
The ‘first order change in utility’ (or ‘total differential’) is:
\[ dU = \frac{\partial U}{\partial X}dX + \frac{\partial U}{\partial Y}dY\] \[ = MU_{X}dX + MU_{Y}dY\]
Essentially, approximates the total change in utility for very small changes in X and Y. Setting it equal to zero and rearranging it yields the rate, at the margin, one is willing to give up Y for X:
\[ dU = MU_{X}dX + MU_{Y}dY = 0 \]
\[\frac{dY}{dX}=-\frac{MU_{X}}{MU_{Y}}\]
Rearranging the utility maximising condition yields more intuition:
\[P_X/P_Y = MRS = MU_X/MU_Y\]
(at each consumption point X,Y)
\[\frac{MU_X}{P_X} = \frac{MU_Y}{P_Y}\]
Same ‘bang for each buck’ (if optimising)
##Caveat on ‘corner solutions’
Advanced: This is a necessary but not sufficient condition, sufficient if DMRS everywhere
But…
But…
Enrichment:
Constrained problem equivalent to optimising a single unconstrained ‘Lagrangian’ function
(subject to non-negativity constraints)
##App 2.4: ticket scalping
##App 2.5: What’s a rich uncle’s promise worth?
##Using the model of choice
Move to ppt slides here beginning with ‘Utility Maximization: A Graphical View’
Consider: are these ‘perfect substitutes’ for someone who wants caffeine, but has no taste buds?
\[U(X,Y)=4X+3Y\]
Rates each increase utility per-unit (derivative) are constant: \(MU_X = 4\), \(MU_Y = 3\)
So (for perfect substitutes) buy the one that increases it more *per-
With perfect substitutes: ‘Bang for the buck’ rule
\[U(X,Y)=4X+3Y\]
Here, if \(4/P_X > 3/P_Y\), then buy X
if \(4/P_X < 3/P_Y\), then buy Y; (if equal, buy either)
Rearranging, if \(P_X < 4/3 P_Y\), buy X … etc.
Warning: If not perfect substitutes, MU ratios depend on consumption levels.
Mathematical function example:
\[U(X,Y)=min(2X,Y)\]
E.g., X: bicycle frames, Y: wheels.
Warning: this min function looks backwards, but it’s correct; see notes
A Cobb-Douglas example
\[ U(X,Y)=\sqrt(XY) \]
\[MU_X = \frac{\partial}{\partial X} (XY)^{1/2} = \frac{1}{2} (Y/X)^{1/2}\]
\[MU_Y = \frac{1}{2} (X/Y)^{1/2}\]
Here, amount of Y you’d give up to get a unit of X:
\[MRS(X,Y)= MU_X/MU_Y = Y/X\]
{Check reasonable: The more Y I’ve , the more Y I’d give up to get another X :)}
\[ U(X,Y)=\sqrt(XY) \]
\[MRS(X,Y)= MU_X/MU_Y = Y/X\]
Calculus:
\(MU_X\) is the slope of \(U(X,Y)\) in X at a particular point, i.e., the (partial) derivative with respect to X
\[MU_X = \frac{\partial}{\partial X} (XY)^{1/2} = \frac{1}{2} (XY)^{-1/2}Y = \frac{1}{2} (Y/X)^{1/2}\]
Similarly,
\[MU_Y = \frac{1}{2} (X/Y)^{1/2}.\]
…Cobb-Douglas ctd
\[MRS(X,Y)= MU_X/MU_Y = Y/X\]
Here utility-maximization requires, at optimal choices of X and Y: \[MRS(X,Y)= Y/X = P_X/P_Y\]
For any price ratio, find ratio of Y & X.
With prices and income, \(I\), find consumption of X & Y.
Rearranging optimization condition:
\[Y P_Y = X P_X\]
Optimization condition for this particular utility function:
\[Y P_Y = X P_X\]
Combining this with the budget constraint \[P_X X + P_Y Y = I\]
solve for X & Y, as fncns of prices & income (see notes) \(\rightarrow\)
\[Y = I/(2P_Y)\] \[X = I/(2P_X)\]
Complicated (e.g., nonlinear) pricing and budget constraints
Many goods; when can I consider ‘all other goods’ as a ‘composite good’ in considering price changes, etc?
###Cobb-Douglas general form (from other text)
\[U(X,Y)=X^aY^b\]
where a, b are positive constants
This is “homothetic”:
\[MU_X=\frac{\partial U}{\partial X} = aX^{a-1}Y^b\] \[MU_Y=\frac{\partial U}{\partial Y} = bX^{a}Y^{b-1}\]
Taking the ratio of these yields
\[MRS = \frac{a}{b}\frac{Y}{X}\]
So only the ratio Y/X affects the MRS; e.g., double both and MRS is the same.
###A non-homothetic example
\[U(X,Y)=X+ln(Y)\]
Here Y has a diminishing MU, but for X MU is constant
\[\frac{MU_X}{MU_Y}= \frac{1}{1/Y}=Y\]
So the MRS increases as Y increases, but it’s independent of the amount of X consumed.
So, if we double both then (class question)?
Ans: MRS doubles
This represents Quasilinear preferences:
More generally (and usefully) \(U=x_1+f(x_2,...,x_n)\) linear in one ‘numeraire’ good \(x_1\).
##App 2.6: Loyalty programmes (Skip in lecture)
Some examples of things you will need to learn and use for a PhD micro module
Another taste, UCLA notes here
(Doctoral-prep concepts)
A formal definition and proof of the idea that a utility function ‘represents’ preferences
Exercises like ‘show that lexicographic preferences cannot be represented by a continuous utility function’
What conditions on a (multivariate) utility function ensure it exhibits the equivalent of a diminishing marginal rates of substitution?
Representation of the MRS between all goods as a matrix, properties of this
Preferences, Utility, Consumer optimization,
individual and market demand curves.
This is a very large problem set (others are smaller).